Gravitational aberration is the counterpart of optical relativistic aberration, but with the use of gravitational rather than EM signals.
Conventional relativistic aberration changes the angles of light-rays, so that a ray that one observer carefully emits at angle [math]A[/math] may be seen by another observer to have a different angle, [math]A'[/math]. The effect makes rays emitted by a moving body appear tilted forwards. For an observer (or camera) moving at speed through a uniform starfield, aberration makes more of the starlight hit their front than their rear, and makes the background starfield seem to be more concentrated in the forward direction then the rearward one.
If gravitational signals have a finite speed, one would expect a similar effect to work for gravitational sources, and of the speed of gravity was exactly c, one would expect exactly the same aberration relationships. The apparent gravitational position of a star would change by exactly the same angle as its optically-viewed position.
If this didn't happen, a star would have two apparent positions depending on whether we used its gravitation or its light to locate it. This would make relativity theory rather more complicated, as a star's apparent position" for a specified observer would no longer be a fundamental property - we would have at least two positions to choose from, which would vary according to at least two different sets of equations.
Apparent absence of gravitational aberration
In most C20th relativity textbooks, a discussion of gravitational aberration is notable for its absence, as gravitational aberration on its own seems to destroy some otherwise nicely-balanced systems, including (allegedly) orbital mechanics.
N1L and gravitational aberration
The simplest example of a system disturbed by g.a. is Newton's First Law, which says that a moving body must continue to move at the same speed indefinitely until something intervenes.
A distorted starfield seen by a fast-moving astronaut will show an apparent concentration of stars in the forward direction – if the gravitational attraction of each of these stars appears to point to its visible position, the astronaut should expect to undergo a free-fall acceleration forwards, towards the region of highest mass-density.
This would then further increase their speed wrt the starfield, further increasing the apparent anisotropy in the stars' positions, and create an even stronger forward pull.
The result would be a deeply unstable universe in which any form of relative motion would be liable to cause a runaway positive-feedback acceleration.
As this is obviously not real-world behaviour in our universe, it was common for theorists to say, when one twisted their arm, that we knew (empirically) that there was no such thing as gravitational aberration, and leave it at that. Stars were instead supposed to have a sort of ventriloquist-like ability to make their gravitational fields appear to originate from their instantaneous positions rather than their apparent, time-legged positions, making it appear as if the speed of gravitation was infinite.
Gravitational aberration and gravitoelectromagnetism
The apparent dismissal of an effect from gravitational theory for pragmatic rather than theoretical reasons did not sit well with everybody, and in 19xxx Steve Carlip produced a study of the question of gravitational aberration, in which he drew on analogies with electromagnetism. Carlip's conclusion was that obvious bulk gravitational aberration effects shouldn't show up in gravitational theory, not because they didn't exist, but because their main results were cancelled out by velocity-dependent gravitoelectromagnetic effects.